Me, I'd want something like the Enterprise-J in size. I don't think it needs to be quite as flimsy looking -- I don't like the really skinny pylons for the warp-nacelles, and I'd probably want three or four engines. I'd like a little more depth to the vessel as well. I'd want to keep the crew fairly small relative to the size and instead use as much of the volume for supplies, equipment, reactors, some fuel and so on.
What I'd do with it? I'd probably want to go back in time and alter the path of history in a way that would be better for the United States and for Democracy in general. I'd want to rein the bankers in a great deal, perhaps prevent the Federal Reserve act from getting passed, perhaps expose the bankers for causing the 1907 banking crisis. I would also want to make the Iron Law of Oligarchy and the Iron Law of Institutions more popular in order to make people more aware as to how corruption arises, and why Communism is not a realistic option.
Once I'd reform history in the way that I like it and see fit, then I'd go find other star-systems that could either be terra-formed, and inhabited.